We give the characterizations of the classes of matrix trans-formations ($m(phi),{ell}_p$), ($n(phi),{ell}_p$) ([5, Theorem 2]), (${ell}_p,m(phi)$) ([5, Theorem 1]) and (${ell}_p,n(phi)$) for $1{leq}p{leq}{infty}$, establish estimates for the norms of the bounded linear operators defined by those matrix transformations and characterize the corresponding subclasses of compact matrix operators.